Casimir squared correction to the standard rotator Hamiltonian for the O(n) sigma-model in the delta-regime

Abstract In a previous paper we found that the isospin susceptibility of the O(n) sigma-model calculated in the standard rotator approximation differs from the next-to-next-to leading order chiral perturbation theory result in terms vanishing like 1/ℓ, for ℓ = L t /L → ∞ and further showed that this deviation could be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions, by Balog and Hegedüs for n = 3, 4 and by Gromov, Kazakov and Vieira for n = 4, and find good agreement in both cases. We also consider the case of 3 dimensions.

Casimir squared correction to the standard rotator Hamiltonian for the O(n) sigma-model in the delta-regime

HJE
Hamiltonian for the O(n) sigma-model in the delta-regime
F. Niedermayer 0 1 2
P. Weisz 0 1
Switzerland 0 1
0 80805 Munich , Germany
1 Institute for Theoretical Physics, University of Bern
2 Albert Einstein Center for Fundamental Physics
In a previous paper we found that the isospin susceptibility of the O(n) sigmamodel calculated in the standard rotator approximation di ers from the next-to-next-to leading order chiral perturbation theory result in terms vanishing like 1=`, for ` = Lt=L ! 1 and further showed that this deviation could be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions, by Balog and Hegedus for n = 3; 4 and by Gromov, Kazakov and Vieira for n = 4, and nd good agreement in both cases. We also consider the case of 3 dimensions.
E ective Field Theories; Lattice Quantum Field Theory; Sigma Models
1 Introduction
2 The isospin susceptibility
3 Delta regime in d = 2
3.1
3.2
3.3 The O(
4
) case
Running coupling functions
The Balog-Hegedus results for the I = 1; 2 energies for O(
3
)
4 The case d = 3
A Spectrum of O(
3
) rotator in d = 1 at nite lattice spacing
B The sunset diagram for D 2; 3; 4 for large `
C The equations for the lowest energy state in a nite volume for the
vacuum, 1- and 2-particle sectors
C.1 The iterative calculation of the vacuum energy
C.2 Using discrete FT C.4 The energy of the 1-particle sector C.5 The ground state energy of the 2-particle sector C.3 N -dependence of the sums appearing in FT and the Euler-Bernoulli relation 20
1
Introduction
In the pioneering paper [1] Leutwyler showed that to lowest order in chiral perturbation
theory ( PT) the low energy dynamics of QCD in the
regime is described by a quantum
rotator for the spatially constant Goldstone modes. We recall that for a system in a periodic
spatial box of sides L the
regime is where the time extent Lt
L and m L is small
(i.e. small or zero quark mass) whereas F L, (F the pion decay constant) is large.
There are other important physical systems, in particular in condensed matter physics
where anti-ferromagnetic layers are described by the O(
3
) sigma-model for d = 3 [2],
where the order parameter of the spontaneous symmetry breaking is an O(
3
) vector. In
the analogous perturbative regime these systems and also the non-linear sigma models in
d = 2 are described by a quantum rotator to leading order.
{ 1 {
In all such systems the lowest energy momentum zero states of isospin I have to leading
order PT energies of the form
where
EI / Cn;I ;
Cn;I = I(I + n
2) ;
(1.1)
(1.2)
is the eigenvalue of the quadratic Casimir for isospin I.
At 1-loop level it turns out that the Casimir scaling (1.1) still holds, but it is of course
expected that at some higher order the standard rotator spectrum will be modi ed. The
standard rotator describes a system where the length of the total magnetization on a
timeslice does not change in time. This is obviously not true in the full e ective model given by
PT. To our knowledge, the actual deviation from the Casimir scaling was rst observed by
Balog and Hegedus [3] in their computation of the spectrum of the d = 2 O(
3
) non-linear
sigma model in a small periodic box (circle) using the thermodynamic Bethe ansatz (TBA).
Of course, a deviation from the standard rotator spectrum could be established by
explicit perturbative computations. However, in our previous paper [5] we pointed out
that by comparing the already obtained NNLO results for the isospin susceptibility
calculated in
PT at large `
Lt=L with that computed using the standard rotator one can
establish, under reasonable assumptions, that the leading correction to the rotator
Hamiltonian occurs at 3-loops and is proportional to the square of the Casimir operator with a
proportionality constant determined by the NNLO LEC's of
PT.
In appendix A we illustrate that such corrections are expected by recalling the spectrum
of the O(
3
) rotator in d = 1 with lattice regularization. Note that the appearance of terms
proportional to CI2 is not a lattice artifact. This simple example serves only to show that
distorting the Lagrangian of the standard 1d rotator by an O(
3
) invariant perturbation
leads naturally to a spectrum which (at a higher order) contains such term.
After reviewing some preliminary results in section 2, in section 3 we test our claim
above in the d = 2 O(
3
) and O(
4
) non-linear sigma models in a periodic box. For O(
3
)
we nd excellent agreement of our prediction with the analytic computations of the lowest
isospin 1 and 2 energies for O(
3
) by Balog and Hegedus [3].
For O(
4
) Balog and Hegedus [4] computed only the ground state energy; Gromov,
Kazakov and Vieira [6] computed also higher state energies but the results at small volumes
presented there are not su ciently precise for our purposes. We thus generated more data;
our methods used to solve the TBA equations are described in appendix C. Again there is
good agreement with our prediction.
Finally in section 4 compute the e ect for the d = 3 O(n) model, but we have not yet
found data for which a comparison can be made.
The derivation of our result depends on the validity of a (plausible) assumption; but
also there is, to our knowledge, no rigorous derivation of the TBA equations (or even the
S-matrix) from rst principles starting with the 2d O(n) QFT.1 Hence the agreement of the
results for d = 2 provides extra evidence for the validity of both scenarios, and furthermore
encourages the application of our assumption for d > 2.
1Except for n = 4 which is also a principal chiral model.
{ 2 {
The isospin susceptibility
In this section we recall some results on the isospin susceptibility. The Hamiltonian of the
O(n) standard quantum rotator with a chemical potential coupled to the generator L^12 of
rotations in the 12-plane is
H0(h) =
hL^12 ;
L^2
2
where
is the moment of inertia. In d = 4 dimensions to lowest order
PT one has
' F 2L3. The isospin susceptibility is de ned as the second derivative of the free energy
w.r.t. h:
HJEP05(218)7
In ref. [5] we showed that for small u = Lt=(2 ) the isospin susceptibility computed from
the standard rotator, which we call rot is given by
1
(2.2)
rot =
Lt
nLd 1u
1
45
1
(n
2)u +
(n
2)(n
4)u2 + : : : :
(2.3)
On the other hand in a previous paper [7], we computed the isospin susceptibility in
an asymmetric Lt
Ld 1 box (with periodic boundary conditions) and the mass gap, in
this case the lowest energy in the isospin 1 channel, to NNLO (next-to next-to leading
order)
PT. For the susceptibility we recall the results in eqs. (3.54){(3.57) of [7] with
dimensional regularization:
with
and
with
(2.1)
(2.4)
(2.5)
(2.6)
(2.7)
2
1
3
=
ng02 1 + g02R1 + g04R2 + : : : ;
R1 =
2(n
2)I21;D ;
R2 = R(a) + R2(b) ;
2
2
R(a) = 2(n
2)
W + 2I21;D I10;D
I21;D +
I31;D
;
2(n
2)
VD
and R2(b) involves the 4-derivative couplings which we shall not include in this paper. The
expressions above require some explanation. Firstly g0 is the bare coupling for d = 2;
0
g 2 =
constant in the chiral limit. VD is the volume VD = LtLD 1`^q where with DR we have
s, the spin sti ness for d = 3; and g0 2 = F 2 for d = 4 where F is the pion decay
added q = D
d extra dimensions with extent L`^; in this paper we will usually set `^ = 1
unless stated otherwise. (Note that the renormalized quantities do not depend on `^.) Inm;D
are 1-loop dimensionally regularized sums over momenta (cf. eq. (3.44) of [7]).2 Finally W
2Note in [7] we dropped the label D and wrote only Inm.
{ 3 {
is discussed in appendix B.
eqs. (5.9){(5.11) of [7]:
(here V D = LD 1`^q), with
4
4
in (2.7) is an integral associated with a two-loop vacuum massless sunset diagram, which
The lowest energy state above the vacuum (the mass gap) is given by E1 = m1 in
E1 =
2V D
n1g02 h1 + g024(2) + g044(
3
) + : : : i ;
Here R(z) is the propagator for an in nitely long strip without the slow modes and W is
a 2-loop sunset integral discussed in appendix B.
For the simple rotator (2.1) at zero isospin chemical potential, the energy gap is
given by
E1(L) =
(n
1)
2 (L)
:
appendix B of ref. [5] for d = 4.
exponentially with `):
By inserting the expression for
using (2.8) into (2.3), we obtain the susceptibility for small
u as a function of F; L; `. This can then be compared to the direct
PT computation (2.4) in the -regime for ` 1. The comparison requires knowledge of the large `-behavior of shape functions and the sunset integral appearing in the latter, which are discussed in
The two results in NNLO di er by / 1=` terms for `
1, (plus terms vanishing
I10:D 1 + (n
3)R(0)2 + 4-derivative terms :
The leading order of the coe cient (L) in (2.13) is directly related to
we considered only the case d = 4 ; in the next sections we consider the lower dimensions
d = 2; 3.
{ 4 {
rot =
1
F 4L4
`
2 + : : : + O
1
F 6L6
;
for d = 4 and similarly for d = 2; 3.
In ref. [5] we showed that the deviation above can be accounted for if the spectrum to
the order we are considering is given by a modi ed rotator with eigenvalues of the form
(L) appearing here contains a higher order correction with respect to the one in (2.11),
denoted below by
old(L).
EI (L) =
1
2 (L) Cn;I +
L
(L) Cn2;I :
1
(L)
=
1
old(L)
2(n
1)
L
(L) :
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
HJEP05(218)7
It is interesting to note that the correction discussed here shows up at NNLO in the
isospin susceptibility
, and in the NNNLO (next order) in the spectrum of the e ective
rotator. The reason is that in the (standard) rotator approximation the Boltzmann weight
is
exp( g2CI Lt=L) = exp( g2`CI ) where g
2 = g2(1=L). The typical values of the
isospin in the partition function are then given by CI
exp( CI2 (L)Lt=L) in the Boltzmann weight gives a correction
=
the susceptibility. With
(L) =
3g8 + O g10 (cf. [5]), the leading / CI2 term in the
1=(g2`), hence the extra factor
(L)=(g4`) for
e ective Hamiltonian which is of NNNLO, results in an NNLO, / g4=` term in
= .
3
Delta regime in d = 2
The susceptibility computed in
PT is for d = 2 given by (cf. eq. (3.67) of [7])
=
2
ng2MS(1=L)
(n
2 n
2) 2(2)(`)
(n
2)
where gMS(q) is the minimal subtraction (MS) scheme running coupling at momentum
scale q. The function r2(`) appearing above is given by (cf. eq. (3.68) of [7])
r2(`) = w(`)
2 10(`)
12 2(2)(`)
where w(`) appears in the expansion of the sunset diagram de ned (see (B.5)).
For large ` (cf. (B.20), (B.25), (B.28)-(B.30) of [5]):
HJEP05(218)7
1(2)(`) ' 1(1=)2(
1
) +
2(2)(`) ' 1(1=)2(
1
)
3(2)(`) ' 3(1=)2(
1
)
3
`
2 +
2
3
+
2 +
` ;
2
3
where gFV is the LWW running coupling [8] de ned through the nite volume mass gap:
Its expansion in terms of the running coupling in the MS scheme of DR is given by (cf.
eq. (2.2) in [8]):
g2FV(L) = g2MS(1=L) + c1g4MS(1=L) + c2g6MS(1=L) +
The rst two coe cients were computed in ref. [8]
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
where
with
Z
ln(
4
)
;
( =
The 3-loop coe cient c3 can be obtained by combining Shin's 3-loop computation [9]
of the
nite volume mass gap using lattice regularization, with the computation of the
3-loop coe cient of the lattice beta-function by Caracciolo and Pelissetto [10].3 The result
(n
2)
where we have used Veretin's precise numerical values [13] of the lattice integrals appearing
in the formulae, and the si are Shin's constants (appearing in eq. (2.25) of [9]) for which
he quotes the following numerical values (in eqs. (2.28){(2.30) of [9]):
the perturbative result given in (3.1). First we see that the g0MS(1=L) terms match for
large ` if
This is satis ed since (using (3.4))
which is a special case of the general relation
(n
2
2) 2(2)(`) ' 2c1 +
(n
3
2)
` :
1(1=)2(
1
) = 2
Z ;
s(
1
)(
1
) =
(
1
)
1=2 s(
1
) =
1
s(1
2s)
+ 2
s (s) (2s) :
Matching at order g2MS(1=L) requires4
w(`)
2 10(`) ' 45
Using eq. (B.11) and the fact that 10(
1
) is nite consistency requires
{ 6 {
3Note that the original article ref. [10] had a few misprints, some of which were noticed by Shin [11]; the
nal correct result was rst presented in ref. [12].
4Note the terms proportional to (n
2) match.
consistency is obtained if
We arrive at the result we would expect with where
On the other hand, from our considerations of the modi ed rotator (cf. eq. (11.5) of [5])
HJEP05(218)7
where
3 is the leading coe cient in the perturbative expansion of (L), assuming the
expansion starting at order g8MS:
Comparing (3.26) with (3.27) determines
As a consequence, the low-lying spectrum to order g8MS is given by
LEI (L) =
1
2 Cn;I g2MS(1=L)n1 + c1g2MS(1=L) + c2g4MS(1=L)
+ c3g6MS(1=L) + : : : o + Cn2;I 3g8MS(1=L) + : : :
Hence we conclude, for example,
computations of the spectrum of excited states exist. The rst computations for O(
3
) were
done by Balog and Hegedus [3] using the TBA equations with the knowledge of the exact
Zamolodchikov S-matrix [14]. Later Gromov, Kazakov and Vieira [6] computed the excited
states for O(
4
) from Hirota dynamics; in earlier papers Balog and Hegedus [4, 15] computed
the mass gap E1 for O(2r) but not E2.
{ 7 {
For the perturbative analysis of their data, Balog and Hegedus [4] introduced a function
g2J(L) of the box size L,5 through
where b0 ; b1 are the universal rst perturbative coe cients of the
function:
1
g2J(L)
+
b0
b1 ln(b0g2J(L)) =
b0 ln( FVL) ;
small.6 We chose the solution which is small for
FVL small, which has the property
For
FVL small there are usually two solutions of (3.34) for g2J(L), one large and one
g2J(L) = g2FV(L) + O g6FV(L) ;
FVL
1 :
It is appropriate to call gJ a \running coupling function" since it satis es the equation
i.e. like a perturbative running coupling with
function coe cients br = b0(b1=b0)r ; 8r.
Balog and Hegedus consider g2J as a function of z = M L where M is the in nite volume
mass gap:
1
De ning J = g2J=(2 ) the equation becomes [4]
with
In particular, 1
J(z)
+ ln( J(z)) =
(n
2) ln(z) + J (n) ;
J (n) = (n
5Analogous constructions can be made for functions of lengths in in nite volume.
6The solution of the equation 1 + ln( ) =
ln(X) is
=
1
W 1( X) , where W 1(z) is the Lambert
W function [16], rst studied by Euler in 1783, and the index 1 refers to the real branch for which
1= ln(X) ! +0 when X ! +0.
7M= MS = (8=e)4= (1 + 4) and FV= MS = exp f Z=2g = MS= MS.
{ 8 {
The LWW coupling has the following expansion in terms of g2J:
gFV = g2J 1 + (n
2
2) J2 + j3 J3 + : : : ;
(3.43)
with the coe cient j3 given by
1
8
j3 =
R2(n
2)3 +
R1
Z2 + 4Z
(n
2)2
1
3
+ [R0 + 6 + 6 (
3
) + 4Z] (n
2)
2
6 (
3
) ;
(3.44)
where the constants Ri are given in (3.13){(3.15). Inserting the numerical values (3.16){
(3.18) we obtain
j3 = 1:195(
4
) ; (n = 3) ;
(3.45)
in agreement with the value quoted in eq. (4.4) of [3].
3.2
The Balog-Hegedus results for the I = 1; 2 energies for O(
3
)
In table 1 we reproduce the data of Balog and Hegedus [3]; (f3;est appearing in the last
column is de ned in (3.48)). Firstly we remark that by tting the data for the mass gap
E1 one obtains a value of j3 close to that given in (3.45) deduced from Shin's analysis.
Secondly, from the fth column of table 1, we see that although the ratio E2=E1 is
close to the ratio of the Casimir eigenvalues, the data of Balog and Hegedus establishes
that the simple e ective rotator model requires corrections.
We have for the I = 2 mass for n = 3 the expansion
LE2(L) = 6
J(z) 1 + J(z)2 + (j3
8f3) J(z)3 + : : : ; (n = 3) :
With our value of f3 we get a very small 3-loop coe cient
(3.46)
(3.47)
j3
8f3 =
0:007(
4
) ; (n = 3) ;
{ 9 {
0.2
0.15
O(
3
), quadratic fit
O(
3
), cubic fit
O(
4
) data
O(
4
), quadratic fit
O(
4
), cubic fit
prediction
0
and cubic ts. The star indicates the prediction in (3.30) for L ! 0 limit.
which is similar to the remark in [3] (before their eq. (4.7)) that they numerically established
that the 3-loop coe cient was zero. Although Shin's numbers (which have not yet been
checked by an independent computation) are consistent with the data, to indicate whether
or not j3
8f3 is non-zero would require the values of the si to a greater (at least a factor
10) accuracy than that obtained by Shin.
To see even more clearly the agreement of our analysis with the data of Balog and
Hegedus, in gure 1 we plot estimates for f3;est given by
f3;est =
1
(n2
1)(n
2)2
4
J
(n
1)
2n
for the case n = 3 (and for n = 4, see next subsection). Comparing the quadratic and cubic
ts to the data in the range 0:05 <
J < 0:089 (the rst 8 points) one gets f3 = 0:151(2),
which is in excellent agreement with our prediction in (3.30).
3.3
The O(4) case
In [4] the 1-particle data for z
0:001 for the O(
4
) model is su ciently precise for our
purposes, however, the 2-particle results are missing. Gromov, Kazakov and Vieira [6]
computed the 2-particle state energies, but for z < 0:1 we required higher precision than those
presented in that reference. For z
0:1 we used the data obtained using the Mathematica
program in ref. [6].8
Afterwards we have rewritten the code, discretizing the continuum formulation and
using Fast Fourier Transform (FFT) to calculate the convolutions appearing. Due to FFT
8We thank Nikolay Gromov for sending these data to us.
0.0005
0.001
1(z) discussed at the end of appendix C.4.
the calculation of an iterative step becomes much faster and yields more precise values. In
addition, approximating the 1d continuum functions by a
nite lattice of N points, and
using several N values (typically N = 210 : : : 214), one can get the continuum limit by
determining numerically the coe cients of the corresponding 1=N expansion. Some details
are given in appendix C.
With this technique we could lower z, the size of periodic box down to z = 0:0005 .
For z < 0:0005 it becomes more time consuming to produce reliable estimates,9 but the
values of J(z) (see table 3) are already quite small.
Our data for the TBA energies EI for I = 0; 1; 2 is presented in table 2. The
corresponding values for the excitation energies
EI (L) = EI (L)
E0(L)
(3.49)
for isospins,10 I = 1; 2 together with
J are given in table 3.
In that table we see that Casimir scaling holds approximately for a large range of
volumes. Also we show estimates for f3;est given by (3.48); which are plotted in
gure 1
together with quadratic and cubic
ts. In the range 0:0437 <
J < 0:075 (7 points)
one gets f3 = 0:151(
3
). Like in the O(
3
) case, this is consistent with our expected value
f3 = 0:150257 in (3.30).
4
The case d = 3
is given, already in [2],
From eqs. (3.54){(3.57) and (3.72){(3.74) of [7] the expansion of the susceptibility with DR
=
2 s
n
1 +
1
1
sL Re1 +
( sL)2 Re2 + : : : ;
(4.1)
9Note that with decreasing z the region of attraction shrinks, and it becomes more and more di cult to
start the iterative method in the corresponding region.
10In the sector with a given isospin I only states with particle number
I contribute. Among these
the
= I is the lightest, and the TBA equations used describe just the = I = 1 and the
= I = 2 states.
HJEP05(218)7
z
We require the large ` behaviors of the shape functions which we take from eqs. (B.20)
and (B.27) of [5]:
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
For the mass gap we have from eqs. (5.9){(5.11) of [7] (setting `^ = 1)
E1 =
(n
1)
2 sL2
1 +
1
sL 4e
(2) +
( sL)2 4e
(
3
) + : : : ;
with
e
4
e
4
where we have set the coe cients of the four-derivative couplings l1 = l2 = 0.
+
1
From eq. (3.16) of [18] for D
3:
So for d = 3
2)
2q1 +
16
3) singularities canceling as they should.
For the rotator we get
rot =
1
and rot agree at 1-loop order up to terms vanishing exponentially
with ` since (see eq. (5.10) of [18])
LD 2R(0) =
u 1=2 S(u)D 1
1 Z 1
0
=
=
4
1
1
4
4
1=2
(D 1)(
1
)
1(2=)2(
1
)
2(D
D
1
1)
2
4
for D = 3 :
(4.10)
(4.11)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(n
2)(n
4)
(4.12)
3 =
1
Nyfeler and Wiese [19] have investigated the histogram of the uniform magnetization
for the spin 1/2 Heisenberg model on the honeycomb lattice in the cylindrical regime (see
eqs. (3.9){(3.12) and gure 4.3). Having previously determined the low-energy constants
we obtain
order ( sL) 4:
which determines
(L) =
X
r=3
r( sL) r 1
;
rot =
4
` 3(n + 1)( sL) 2 + : : :
Then one can check, using eqs. (4.4){(4.6) and (B.15), that at 2-loop
and rot di er
only by terms which vanish as ` ! 1, and neglecting terms which vanish exponentially
rot =
(n
Suppose the spectrum is given by (2.13) and that (L) has the expansion starting at
then from our considerations of the modi ed rotator (cf. eq. (11.5) of [5])
s and c for this model in the cubical regime, they took the NLO formula for
and the
standard rotator spectrum, to compare the cylindrical regime. The agreement seemed to
be very good, but their statistical errors are too large to see signs of our predicted Casimir2
term (or even the NNLO term in
). We are not aware of any other measurements of the
spectrum of the d = 3 O(n) models.
Acknowledgments
correspondence.
We would like to thank Janos Balog, Nikolay Gromov and Uwe-Jens Wiese for useful
The aim here is to illustrate that the energy levels at a given order of the lattice spacing
are polynomials of CI = I(I + 1), not only of I.
The eigenvalues of the transfer matrix are given by
where PI (z) are the Legendre polynomials. These are polynomials in 1 z with coe cients
which are polynomials of CI of order I:
I ( ) =
tially small terms / e 1= , one obtains the expansion11
In the continuum limit one should take
! 0. Neglecting non-perturbative,
exponen(A.1)
(A.2)
(A.3)
For the energies one obtains
1
EI ( ) =
ln ( I ( ))
= CI (1 + )
1
3 CI (CI
6) 2 +
Here the coe cients of n are indeed polynomials in CI .
11Note that the summand vanishes for n > I hence the upper limit of the sum could be extended to
in nity.
The sunset diagram for D
The sunset diagram for the susceptibility is (cf. [18] (4.1) and (4.36))
(`; `^) =
L2D 4W L2D 4
dx G(x)G2(x)
Z
VD
10 g(0; `; `^)
=
1
48 2(D
4)
1
VD
1
16 2 W(`) + O (D
4) ;
here we have reinstated `^, but since we are working with `^ = 1 we have VD = ``^D 4 = `.
For the analogous diagram in the in nite strip one had (cf. eqs. (5.11) and (5.61)
1 one has up to exponentially small corrections (cf. eq. (5.20) in [18])
g(0; `; `^)
1
VD ' R(0; `^) :
In appendix (B.4) of ref. [5] we derived the relation
`2 +
112 1(
3
)(
1
) ` +
9
2
+
1
For the analogous diagram in the in nite strip
Using similar methods to that used to derive (B.4) we obtain for D
2:
B.1
For D
with
1
16 2 W(`)
(`) =
(`)
`
2
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
Comparing this with (B.5) and (B.8) requires
and
w(`) p1 '
with values given in eq. (4.45) of ref. [18]
(`) =
These coe cients are simply related to the coe cients in (4.7) through
Using eq. (4.11) and the numerical values
1 =
2 =
1(2=)2 = 0:0997350799980441171545246633952 ;
1(2) = 0:100879698927913998245389274826 ;
12There is a printing error in eq. (4.44) of ref. [18]; for d = 3, = L2W .
For D
3
has a singularity:
W(`) =
Again using similar methods to that used to obtain (B.4), we can derive the expansion of
for D
3 for large `, and nd that the expansion of (`) large ` is given by:
(`) '
+ q1
32
3 h 1(2)(
1
) 1
i
` h 1(2=)2(
1
) 4i +
3
For the purposes of this paper we do not need the value of the constant q1 appearing in
eqs. (B.14) and (B.15). However we can easily obtain its numerical value from the lattice
computation of the perturbative coe cients of the moment of inertia given in eq. (7.7)
of [20]:
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
(B.15)
(B.16)
(B.17)
(B.18)
(B.19)
(B.20)
Combining this with eq. (B.17) we get
2 = [2q1 + 0:149993826254627007174930112530] (n
2) :
As a check, using eqs. (B.19) and (B.20) and
3(2=)2(
1
) = 0:104412211554310173598670103056 ;
we obtain, by comparing (B.15) with (B.13) at ` = 2, the estimate q1
0:07521075 which
agrees with (B.22) to nearly 5 digits.13
C
The equations for the lowest energy state in a
nite volume for the
vacuum, 1- and 2-particle sectors
We consider here the SU(2) SU(2) principal chiral model which is equivalent to the O(
4
)
non-linear sigma model.
Al. and A. Zamolodchikov [14] proposed a self consistent exact S-matrix for the
principal chiral models, which is built from the factors
S0(x) = i
(1 + ix=2) (1=2
ix=2)
(1 ix=2) (1=2 + ix=2)
:
In the equations for I = 0; 1; 2 discussed below the following kernels occur.
Sp(x) = S0(x + i=2) ;
K0(x) =
ln S0(x)2
1 d
2 i dx
1
2
Km(x) = K0(x i=2) ;
Kpp(x) = K0(x + i) :
=
[ (1 + ix=2) + (1 ix=2)
(1=2 + ix=2)
(1=2 ix=2)] ;
(B.21)
(B.22)
(B.23)
In the relations above it is assumed that x is real. Unlike the others, Kpp(x) has a pole
term
i=( x). Further, it satis es the relation
Kpp(x) =
K0(x)
1
(1 ix)
+
i
x
:
Below we shall consider separately the cases of the ground states in the sectors I = 0; 1; 2.
The equations to be solved are clearly presented in ref. [4]; our discussion, however is
based on the notations of ref. [6].
13For d = 4 the analogous estimate for cw at ` = 2 was correct to 7 digits.
The iterative calculation of the vacuum energy
The nite-volume vacuum energy is de ned through a single function A(x), which is
obtained by an iterative solution.
The initial function is chosen for convenience as
A(0)(x) = A00(x)
exp ( z cosh( x)) :
The iterative step with the input function A(k)(x) calculates the output function14 A^(k)(x)
through the steps A(k)(x) ! r(x) ! f (x) ! A^(k)(x). The auxiliary functions15 are de ned
below. The reason for using A^(k) for the output, is that the input of the next iteration,
A(k+1)(x), will be modi ed to improve the convergence of the iterations.
and rc(x) = r(x) , the function f (x) is given by
Here ` ' denotes convolution. Due to the singularity of Kpp(x) at x = 0 a Principal Value
integral appears in (C.9) instead of an ordinary convolution, which is more di cult to deal
with numerically.
One can circumvent this problem by introducing the function
(x) =
i
1
x + i
+
x
1
i
;
which besides of the PV integral includes also the extra
rc(x) in (C.9). One obtains then
a simpler relation with two convolutions only,
f (x) = (K0 (r + rc)) (x) + (
rc) (x) :
(C.7)
(C.8)
(C.9)
(C.10)
(C.11)
(C.12)
(C.13)
(C.14)
Finally, the output function of the iteration step is
The vacuum energy after k-th iteration is given by
A^(k)(x) = A00(x) exp f (k)(x) :
E0
(k) =
M
Z 1
1
dx Re(r(k)(x)) cosh( x) :
To improve convergence one can use the well known trick which takes the input for
the next iteration a linear combination
when they appear in the nal answer.
14We denote the output of the iteration with A^ since it could be di erent from the input of the next
15For intermediate quantities like r(x), f (x) we don't write the iteration number k, except for clarity
with a parameter 0 <
1. Note that the choice of the parameter
does not in uence
the xed point (provided that the iteration converges), however, it a ects the properties of
the iteration step, like the speed of convergence or whether the initial function A(0)(x) is in
the domain of attraction of the iterative process. One expects that increasing
increases
the speed of convergence to the
xed point but decreases the convergence region. The
latter could be crucial for small L (i.e. small z) since decreasing L also shrinks the domain
of attraction.
For later use, note that the FT of (x) is quite simple,
~ (p) =
2 ( (p) + ep ( p)) ;
(C.15)
(p) is the Heaviside step function. The function ~ (p) is continuous, but its rst
derivative has a nite jump at p = 0.
C.2
Using discrete FT
The discrete FT in Maple is de ned as
and its inverse
Here
gF [k] = FT(g)[k] = p
g[j] = IFT(gF )[j] = p
1
1
N j=0
N k=0
N 1
X e i 2 kj=N g[j] ;
N 1
X ei 2 kj=N gF [k] :
1) ;
=
dp
2
dp
2
1
p
Q
N
;
dx
=
dp
2
1
N
;
dx = p
Q
N
;
kj
N
xj = jdx ; (j = 0; : : : ; N
pk = k
; (k = 0; : : : ; N
1)
exp 2 i
= exp(ipkxj ) ;
g(xj ) = g[j] ;
g~(pk)
Q gF [k] :
(C.16)
(C.17)
(C.18)
given by X = N dx=2 /
cuto limit.
p
It is assumed that g(x) is a periodic function with period 2X = N dx. Since the
functions appearing here are decreasing fast for jxj ! 1, for su ciently large N one can
consider the function to be periodic, g( X) = g(X) = 0. To represent both the positive
and negative x values in an array with indices 1
j
N , we choose N points with
coordinates [xj ; j = 1; : : : ; N ] = [0; 1; 2; : : : ; N=2
1; N=2; : : : ; 2; 1]dx.
here related to rapidity) and in p-space are proportional to p
For N
! 1 the resolutions 1=dx and 1=dp both in the original x-space (which is
N . The (rapidity) cuto is
N . Therefore the N behavior given in (C.18) for constant Q
obeys both requirements: for N ! 1 one approaches the continuum limit and the in nite
At this point Q is independent on N but otherwise arbitrary. Its value can be chosen to
represent A(x) and A~(p) by their corresponding discretizations equally well. The number
of points in the interval
x, the width of the function jA(x)j, is
x=dx
p
N x=Q.
p
2
x= p
.
Similarly, for A~(p) the number of points in its relevant region in Fourier space is
p=dp
N pQ=(2 ). The value of Q is expected to be optimal when these two numbers are
Z
Further for the convolution one has
(g h)(x) =
dy g(x
y) h(y) =
Z dp
2
g~(p) h~(p)eipx
Q IFT(gF hF )(x) :
(C.19)
Rewriting (C.11) by performing a discrete Fourier transformation one gets
(The factor Q appears only in the rst term!)
C.3
N -dependence of the sums appearing in FT and the Euler-Bernoulli
reAssuming that the function g(x) is smooth for x 6= 0, and it has a jump in the rst
derivative, according to the Euler-Bernoulli relation one has
f (x) = Q IFT (K0F (p)(rF (p) + rcF (p))) (x)
2 IFT ([ (p) + ep ( p)] rcF (p)) (x) :
(C.20)
where g(n)(0) = g(n)(+0)
g(n)( 0) is the jump of the corresponding derivative at x = 0.
Here a is the lattice spacing representing dx or dp.
In our case a2
/ 1=N ; hence under the assumption mentioned above, the corresponding
sums have a 1=N expansion with integer powers. One can calculate a quantity for several
N values and determine from these the expansion coe cients.16 The corresponding ts
produce a very stable continuum extrapolation, often to
12 digits.
It seems that it is better to use the naive non-improved sum, (appearing anyhow in
the FT) whose 1=N behavior is known rather than an improved sum which has a more
complicated (or even unknown) type of approach to the continuum limit. In [6] the authors
The energy of the 1-particle sector
took
C.4
= 1=2.
we had in (C.8) { (C.11).
by a factor of
10 5.
For the 1-particle sector the input of an iteration step consists of a function A(k)(x) and
a number (k). Within an iteration one has the chain [A(k)(x); (k)] ! r(x) ! f (x) !
[A^(k)(x); ^(k)] (cf. [6]). The A(k)(x) ! r(x) ! f (x) part is given by the same expressions
16Allowing in the t odd powers of 1=N 1=2 it turns out indeed that the coe cient of 1=N 3=2 is suppressed
In [6] the authors choose a general (0) as a starting point of iterations. However, one
can verify that (k)
of the particle [4].17
! 0 for k ! 1. The physical reason is that is related to the rapidity
The state in the I = 1 sector with the smallest energy is the 1-particle state with zero
momentum, corresponding to
= 0. Starting the iteration with (0) = 0 all the subsequent
(k) values remain automatically zero. For this reason, we set
= 0, simplifying the
iteration steps to A(k)(x) ! A^(k)(x).
The output function after one iteration (for
= 0) is written in the form
A^(k)(x) = A00(x) [Sp(x)]2 exp(f (k)(x)) ;
(C.22)
(cf. (C.2)). The iteration A(k)(x) ! A^(k)(x) has the same form as for the vacuum, given
by (C.8){(C.12). (Note that due to the factor [Sp(x)]2 the agreement is only formal.)
Introducing the averaging in this case, one obtains for the next input function
dx Re r(k)(x) cosh( x) :
The energy of the ground state in the 1-particle sector for
= 0 after the k-th iteration
The output value of is given by the positive solution of (C.25),
17In the expression of the energy [6] for general the rst term in (C.24) is replaced by M cosh( ). Note
that the second term will also depend on
through r(k)(x) = r(k)(x; ).
1
2
C.5
The ground state energy of the 2-particle sector
For the case of the 2-particle sector, the input of an iteration step consists of a function
A(k)(x) and two numbers, 1(k) and
2(k). In this case again it is enough to restrict the
iteration to the zero total momentum, i.e. 1 + 2 = 0. (The iteration would drive the
system anyhow to satisfy this condition at the FP.)
In the rest frame one has 2 =
1 = =2, where
is related to the rapidity di erence.
Because of the symmetry we can restrict the discussion to > 0.
In the case of zero total momentum the input of the iteration is [A(k)(x); (k)] and the
output is [A^(k)(x); ^(k)].
Consider the equation
z sinh
+ Im ln S02( ) + 2
1
2
= 0 ;
where we introduced the function (depending implicitly on the input variables)
(x) = (Km
r)(x) :
^(k) =
> 0 :
(C.23)
(C.24)
(C.25)
(C.26)
(C.27)
The output function A^(k)(x) after the k-th iteration is given by
A^(k)(x) = A00(x) Sp x +
2
1 (k) Sp x
1 (k)
2
2
exp(f (k)(x)) ;
where the procedure to obtain f (k)(x) is formally the same as for the
= 0; 1 cases. Here
one can introduce two averaging parameters
A and
:
and
The presence of these two parameters plays an important role in the stability of the iteration
and for the speed of convergence. For example, choosing
1 the value of
changes. In our O(
4
) simulations at small z we found that it is much easier to
good starting point, (one in the domain of attraction of the xed point) when one chooses
A. For our last point, z = LM = 0:0005 we took
A = 0:4,
= 0:14.
The energy of the ground state in the 2-particle sector after the k-th iteration is
E2
(k) = 2M cosh
1
2
M
Z
dx Re r(k)(x) cosh( x) :
(C.31)
error smaller than 10 3),
For a given z and N we let the iteration run until the result stabilized (apart from
uctuations due to nite precision) yielding the FP value (
1
)(N ; z) = limk!1
for several N values. (We used N = 210 ; : : : ; 214). With these one can calculate the rst
(k)(N ; z)
few coe cients of the 1=N expansion, (
1
)(N ; z) = (z) + t1(z)=N + t2(z)=N 2 + : : :. This
gives
= (z) = 2(z)
1(z) appearing in (C.25) and tabulated in table 2.
From the numerical analysis described above we found that (z) has a logarithmic
dependence for small z, (z)
ln(z)= + a, where a ' 1:0.
Using the running coupling g2J(z) (cf. (3.34), (3.40)) one obtains a good t (with an
(z)
1
As mentioned earlier, the domain of attraction seems to shrink with decreasing z,
therefore it is important to choose a proper starting value
(0) for the iteration. The
value obtained by (C.32) can be used for (0). Note, however, that we did not investigate
systematically questions related to the domain of attraction.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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